# change of order of integration

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##### 1: Bibliography L
• E. R. Love (1970) Changing the order of integration. J. Austral. Math. Soc. 11, pp. 421–432.
• E. R. Love (1972a) Addendum to: “Changing the order of integration. J. Austral. Math. Soc. 14, pp. 383–384.
##### 4: 2.6 Distributional Methods
Motivated by Watson’s lemma (§2.3(ii)), we substitute (2.6.2) in (2.6.1), and integrate term by term. …Inserting (2.6.2) into (2.6.1) and integrating formally term-by-term, we obtain … The Stieltjes transform of $f(t)$ is defined by … On substituting (2.6.15) into (2.6.26) and interchanging the order of integration, the right-hand side of (2.6.26) becomes … In terms of the convolution product
##### 5: 1.17 Integral and Series Representations of the Dirac Delta
Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): … In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^{2}$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. … Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …
##### 6: 2.10 Sums and Sequences
Assume that $a,m$, and $n$ are integers such that $n>a$, $m>0$, and $f^{(2m)}(x)$ is absolutely integrable over $[a,n]$. Then … This identity can be used to find asymptotic approximations for large $n$ when the factor $v_{j}$ changes slowly with $j$, and $u_{j}$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). …
• (b´)

On the circle $|z|=r$, the function $f(z)-g(z)$ has a finite number of singularities, and at each singularity $z_{j}$, say,

2.10.30 $f(z)-g(z)=O\left((z-z_{j})^{\sigma_{j}-1}\right),$ $z\to z_{j}$,

where $\sigma_{j}$ is a positive constant.

• In these circumstances the integrals in (2.10.28) are integrable by parts $m$ times, yielding …
##### 8: Philip J. Davis
An interesting anecdote told by Davis reveals that he and mathematician Philip Rabinowitz were dubbed “Heroes of the SEAC” when their Gaussian integration code executed correctly on its first run. … Davis also co-authored a second Chapter, “Numerical Interpolation, Differentiation, and Integration” with Ivan Polonsky. … Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. … The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling. …
##### 9: 2.5 Mellin Transform Methods
###### §2.5(i) Introduction
We now apply (2.5.5) with $\max(0,-2\nu), and then translate the integration contour to the right. … Let $f(t)$ and $h(t)$ be locally integrable on $(0,\infty)$ and …Also, let … (The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) …
##### 10: 2.3 Integrals of a Real Variable
###### §2.3(i) Integration by Parts
Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … derives from the neighborhood of the minimum of $p(t)$ in the integration range. … A uniform approximation can be constructed by quadratic change of integration variable: … We replace the limit $\kappa$ by $\infty$ and integrate term-by-term: …